The Gouy phase, sometimes called the phase anomaly, is the remarkable effect that in the region of focus a converging wave field undergoes a rapid phase change by an amount of π, compared to the phase of a plane wave of the same frequency. This phenomenon plays a crucial role in any application where fields are focused, such as optical coherence tomography, mode selection in laser resonators, and interference microscopy. However, when the field is spatially partially coherent, as is often the case, its phase is a random quantity. When such a field is focused, the Gouy phase is therefore undefined. The correlation properties of partially coherent fields are described by their so-called spectral degree of coherence. We demonstrate that this coherence function does exhibit a generalized Gouy phase. Its precise behavior in the focal region depends on the transverse coherence length. We show that this effect influences the fringe spacing in interference experiments in a nontrivial manner. © 2012 Optical Society of America OCIS codes: 030. 1640, 050.1960, 180.3170, 120.3940. In 1890 Gouy found that the phase of a monochromatic, diffracted converging wave, compared to that of a plane wave of the same frequency, undergoes a rapid change of 180°near the geometric focus [1][2][3][4] Under many practical circumstances, light is not monochromatic, but rather partially coherent. Examples are light that is produced by a multimode laser or light that has traveled through the atmosphere or biological tissue. In those cases, the phase of the wave field is a random quantity. Therefore, when a partially coherent field is focused (as described in [13][14][15][16][17][18][19][20]), the Gouy phase is undefined; i.e., it has no physical meaning. In the space-frequency domain, a partially coherent optical field is characterized by two-point correlation functions, such as the cross-spectral density or its normalized version, the spectral degree of coherence [21]. These complex-valued functions have a phase that is typically welldefined. As we will demonstrate for a broad class of partially coherent fields, the phase of both correlation functions shows a generalized phase anomaly, which reduces to the classical Gouy phase in the coherent limit. Furthermore, this generalized Gouy phase affects the interference of highly focused fields, in microscopy for example, altering the fringe spacing compared to that of a coherent field.Consider first a converging, monochromatic field of frequency ω that is exiting a circular aperture with radius a in a plane screen (see Fig. 1). The origin O of the coordinate system coincides with the geometrical focus. The amplitude of the field is U 0 r 0 ; ω, r 0 being the position vector of a point Q in the aperture. The field at a point Pr in the focal region is, according to the Huygens-Fresnel principle [4, Chap. 8.2], given by the following expression:where the integration extends over the spherical wave front S that fills the aperture, s jr − r 0 j denotes the distance QP, and k 2π∕λ is the wavenumber a...

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The phase behavior of Airy beams is studied, and their Gouy phase is defined. Analytic expressions for the idealized, infinite-energy type beam are derived. They are shown to be excellent approximations for finite-energy beams generated under typical experimental conditions. © 2011 Optical Society of America OCIS codes: 050.1960, 350.5030. Beams that do not spread on propagation, so-called nondiffracting beams, have attracted considerable attention since they were discovered by Durnin et al. [1][2][3]. A special type of such beams are the Airy beams described by Berry and Balazs in the context of quantum mechanics [4]. These beams have the remarkable property that they "accelerate" away from the original direction of propagation. Airy beams are idealizations, because they carry an infinite amount of energy. Siviloglou and Christodoulides discussed how an exponentially modulated Airy function source would produce a finite-energy beam, which would retain its nondiffracting and accelerating behavior over an appreciable propagation distance [5]. After the experimental realization of such a beam [6], several studies have been devoted to their properties [7][8][9][10], and a number of applications are being pursued. For instance, the "self-healing" capacity of Airy beams [11] makes them excellent candidates for optical communication through turbulent media [12]. Other intriguing applications are the generation of curved plasma channels [13], and the manipulation of particles along bends in labs-on-achip [14]. Traditionally, the term Gouy phase describes how the phase of a monochromatic, focused field differs from that of a plane wave with the same frequency (see [15] and the references therein). Recently, however, it has also been used to describe the phase of a nondiffracting Bessel beam [16]. In this Letter we study the phase behavior of both finite-energy and infinite-energy Airy beams. By comparing their phase to that of a suitable reference field, their Gouy phase can be defined. A good understanding of the phase properties of Airy beams is of importance in interferometric or remote-sensing applications employing them.Consider a monochromatic, one-dimensional beamlike wave field Uðx; z; ωÞ that propagates in the positive zdirection, and can be written as Uðx; z; ωÞ ¼ ϕðx; zÞe iðkz−ωtÞ ;with the envelope ϕðx; zÞ a solution of the paraxial wave equationHere k ¼ ω=c is the wavenumber associated with frequency ω, c denotes the speed of light, and t the time.A possible solution of Eq. (2) is the so-called Airy beam, given by the expression [4] ϕðs; ξÞwith Ai the Airy function, s ¼ x=x 0 a dimensionless transverse coordinate, and ξ ¼ z=kx 2 0 a normalized propagation distance. In the remainder the constant x 0 is taken to be positive, and the time-dependent part of the wave field is suppressed. An example of the intensity distribution of an Airy beam is shown in Fig. 1, from which both the diffraction-free propagation and the transverse acceleration can be seen.Because of its curved trajectory, we define the Gouy p...

We study the correlation of intensity fluctuations in random electromagnetic beams, the so-called Hanbury Brown-Twiss effect (HBT). We show that not just the state of coherence of the source, but also its state of polarization has a strong influence on the far-zone correlations. Different types of sources are found to have different upper bounds for the normalized HBT coefficient.

The probability density analysis formula of the single orbital angular momentum (OAM) mode of an annular vortex beam with a focusing mirror in a turbulent atmosphere is derived theoretically, and the effects of different parameters on the OAM spectrum are investigated numerically, and the results show that the OAM diffusion of the annular vortex beam is weaker than the Gaussian vortex beam in a turbulent atmosphere under the same conditions, and the annular vortex beam with a focusing mirror can more effectively reduce the crosstalk of OAM modes. Our findings will be useful for improving FSO system performance.